CATEGORIFICATIONOFPRE ...CATEGORIFICATIONOFPRE-LIEALGEBRAS 271 The paper is organized as follows. In Section 2, we recall Lie 2-algebras and their representations, pre-Lie algebras

Theory and Applications of Categories, Vol. 34, No. 11, 2019, pp. 269–294.

CATEGORIFICATION OF PRE-LIE ALGEBRAS AND SOLUTIONSOF 2-GRADED CLASSICAL YANG-BAXTER EQUATIONS

YUNHE SHENG

Abstract. In this paper, we introduce the notion of a pre-Lie 2-algebra, which is thecategorification of a pre-Lie algebra. We prove that the category of pre-Lie 2-algebrasand the category of 2-term pre-Lie∞-algebras are equivalent. We classify skeletal pre-Lie 2-algebras by the third cohomology group of a pre-Lie algebra. We prove thatcrossed modules of pre-Lie algebras are in one-to-one correspondence with strict pre-Lie 2-algebras. O-operators on Lie 2-algebras are introduced, which can be used toconstruct pre-Lie 2-algebras. As an application, we give solutions of 2-graded classicalYang-Baxter equations in some semidirect product Lie 2-algebras.

1. IntroductionPre-Lie algebras (or left-symmetric algebras, Vinberg algebras, and etc.) arose fromthe study of affine manifolds and affine Lie groups, convex homogeneous cones and de-formations of associative algebras. They appeared in many fields in mathematics andmathematical physics (see the survey article [9] and the references therein). The beautyof a pre-Lie algebra is that the commutator gives rise to a Lie algebra and the left multi-plication gives rise to a representation of the commutator Lie algebra. So pre-Lie algebrasnaturally play important roles in the study involving the representations of Lie algebrason the underlying spaces of the Lie algebras themselves or their dual spaces. For example,they are the underlying algebraic structures of the non-abelian phase spaces of Lie algebras[6, 18], which lead to a bialgebra theory of pre-Lie algebras [7]. They are also regarded asthe algebraic structures “behind” the classical Yang-Baxter equations (CYBE) and theyprovide a construction of solutions of CYBE in certain semidirect product Lie algebras(that is, over the “double” spaces) induced by pre-Lie algebras [5, 19]. Furthermore, pre-Lie algebras are also regarded as the underlying algebraic structures of symplectic Liealgebras [13], which coincides with Drinfeld’s observation of the correspondence betweeninvertible (skew-symmetric) classical r-matrices and symplectic forms on Lie algebras [14].In [11], the authors studied pre-Lie algebras using the theory of operads, and introducedthe notion of a pre-Lie∞-algebra. The author also proved that the PreLie operad is Koszul.The PreLie operad is further studied in [10] recently. Furthermore, the relation betweenpre-Lie algebras, trees and cohomology operations are studied in [22].

Received by the editors 2018-03-01 and, in final form, 2019-04-02.Transmitted by James Stasheff. Published on 2019-04-05.2010 Mathematics Subject Classification: 17B99, 55U15.Key words and phrases: 2-algebras, pre-Lie∞-algebras, Lie 2-algebras, O-operators, 2-graded classical

Yang-Baxter equations.c© Yunhe Sheng, 2019. Permission to copy for private use granted.

269

270 YUNHE SHENG

O-operators on a Lie algebra g associated to a representation (V ; ρ) were introducedin [19] inspired by the study of the operator form of the CYBE. See [24] for more details.On one hand, an O-operator could give rise to a pre-Lie algebra structure on V . On theother hand, an O-operator could give rise to a solution of the CYBE in the semidirectproduct Lie algebra gnρ∗ V .

Recently, people have paid more attention to higher categorical structures with mo-tivations from string theory. One way to provide higher categorical structures is bycategorifying existing mathematical concepts. One of the simplest higher structure is a2-vector space, which is a categorified vector space. If we further put Lie algebra struc-tures on 2-vector spaces, then we obtain Lie 2-algebras [1]. The Jacobi identity is replacedby a natural transformation, called the Jacobiator, which also satisfies some coherencelaws of its own. It is well-known that the category of Lie 2-algebras is equivalent to thecategory of 2-term L∞-algebras [1]. The concept of an L∞-algebra (sometimes called astrongly homotopy (sh) Lie algebra) was originally introduced in [21, 26] as a model for“Lie algebras that satisfy Jacobi identity up to all higher homotopies”. The structureof a Lie 2-algebra appears in many areas such as string theory [3, 4], higher symplecticgeometry [2], and Courant algebroids [27].

The first aim of this paper is to categorify the relation between O-operators, pre-Liealgebras and Lie algebras. We introduce the notion of an O-operator on a Lie 2-algebraassociated to a representation and the notion of a pre-Lie 2-algebra, and establish thefollowing commutative diagram:

O-operators on Lie 2-algebras // pre-Lie 2-algebras // Lie 2-algebras

O-operators on Lie algebrascategorificationOO

// pre-Lie algebrascategorificationOO

// Lie algebras.categorificationOO

In [8], the authors introduced the notion of an L∞[l, k]-bialgebra. In particular, anL∞[0, 1]-bialgebra is a Lie 2-bialgebra, which is a certain categorification of the conceptof a Lie bialgebra. See [12, 17, 23] for more details along this direction. However, therelation between Lie 2-algebras and Khovanov-Lauda’s famous work about categorificationof quantum groups [16] is still unclear. 2-graded classical Yang-Baxter equations wereestablished in [8], whose solutions could naturally generate examples of Lie 2-bialgebras.

The second aim of this paper is to construct solutions of the 2-graded CYBE. Wecategorify the relation between O-operators and solutions of the CYBE, and establish thefollowing commutative diagram:

O-operators on Lie 2-algebras // solutions of 2-graded CYBE // Lie 2-bialgebras

O-operators on Lie algebras //

categorificationOO

solutions of CYBEcategorification

OO

// Lie bialgebras.categorification

OO

We also find that there are pre-Lie 2-algebras behind the construction of Lie 2-bialgebrasin [8].

CATEGORIFICATION OF PRE-LIE ALGEBRAS 271

The paper is organized as follows. In Section 2, we recall Lie 2-algebras and theirrepresentations, pre-Lie algebras and their cohomologies, O-operators and solutions ofthe CYBE. In Section 3, first we prove that a 2-term pre-Lie∞-algebra gives rise to aLie 2-algebra with a natural representation on itself. Then we introduce the notion ofa pre-Lie 2-algebra. At last, we prove that the category of 2-term pre-Lie∞-algebrasand the category of pre-Lie 2-algebras are equivalent (Theorem 3.13). In Section 4, westudy skeletal pre-Lie 2-algebras and strict pre-Lie 2-algebras in detail. Skeletal pre-Lie 2-algebras are classified by the third cohomology group (Theorem 4.1). We find that thereis a natural 3-cocycle associated to a pre-Lie algebra with a skew-symmetric invariantbilinear form. By this fact, we construct a natural example of skeletal pre-Lie 2-algebrasassociated to a pre-Lie algebra with a skew-symmetric invariant bilinear form. We alsointroduce the notion of crossed modules of pre-Lie algebras and prove that there is aone-to-one correspondence between crossed modules of pre-Lie algebras and strict pre-Lie2-algebras (Theorem 4.8). In Section 5, we introduce the notion of an O-operator on aLie 2-algebra G associated to a representation (V ; ρ), and construct a pre-Lie 2-algebrastructure on V . In Section 6, we construct solutions of the 2-graded CYBE in the strictLie 2-algebra G nρ∗ V∗ using O-operators (Theorem 6.3). In particular, if the strict Lie2-algebra under consideration is given by a strict pre-Lie 2-algebra, there is a naturalsolution of the 2-graded CYBE in the strict Lie 2-algebra G(A)n(L∗

0,L∗1)A∗ (Theorem 6.4).

At last, we give the pre-Lie 2-algebra structure behind the construction of Lie 2-bialgebrasin [8].

2. Preliminaries2.1. Lie 2-algebras and 2-term L∞-algebras. Vector spaces can be categorifiedto 2-vector spaces. A good introduction for this subject is [1]. Let Vect be the categoryof vector spaces. A 2-vector space is a category in the category Vect. Thus, a 2-vectorspace C is a category with a vector space of objects C0 and a vector space of morphismsC1, such that all the structure maps are linear. Let s, t : C1 −→ C0 be the source andtarget maps respectively. Let ·v be the composition of morphisms.

It is well known that the category of 2-vector spaces is equivalent to the category of 2-term complexes of vector spaces. Roughly speaking, given a 2-vector space C, Ker(s) t−→C0 is a 2-term complex. Conversely, any 2-term complex of vector spaces V : V1

d−→ V0gives rise to a 2-vector space of which the set of objects is V0, the set of morphisms isV0 ⊕ V1, the source map s and the target map t are given by

s(u+m) = u, t(u+m) = u+ dm, ∀u, v ∈ V0,m ∈ V1.

The composition of morphisms is given by

(u+m) ·v (v + n) = (u+m+ n), ∀u, v ∈ V0,m, n ∈ V1, safisfying v = u+ dm.

We denote the 2-vector space associated to the 2-term complex of vector spaces V : V1d−→

272 YUNHE SHENG

V0 by V:

V =V1 := V0 ⊕ V1

syyt

V0 := V0.

(1)

In this paper, we always assume that a 2-vector space is of the above form. Theidentity-assigning map 1 : V0 −→ V1 is given by 1u = (u, 0), for any u ∈ V0.

2.2. Definition. [1] A Lie 2-algebra is a 2-vector space C equipped with

• a skew-symmetric bilinear functor, the bracket, J·, ·K : C × C −→ C,

• a skew-symmetric trilinear natural isomorphism, the Jacobiator,

Jx,y,z : JJx, yK , zK −→ Jx, Jy, zKK + JJx, zK , yK ,

such that the following Jacobiator identity is satisfied,

JJw,xK,y,z ·v (JJw,x,z, yK + 1) ·v (Jw,Jx,zK,y + JJw,zK,x,y + Jw,x,Jy,zK)= JJw,x,y, zK ·v (JJw,yK,x,z + Jw,Jx,yK,z) ·v (JJw,y,z, xK + 1) ·v (Jw, Jx,y,zK + 1).

2.3. Definition. A 2-term L∞-algebra structure on a graded vector space G = g0 ⊕ g1consists of the following data:

• a linear map d : g1 −→ g0,

• a skew-symmetric bilinear map l2 : gi × gj −→ gi+j, 0 ≤ i+ j ≤ 1,

• a skew-symmetric trilinear map l3 : ∧3g0 −→ g1,

such that for any xi, x, y, z ∈ g0 and m,n ∈ g1, the following equalities are satisfied:

(i) dl2(x,m) = l2(x, dm), l2(dm,n) = l2(m, dn),

(ii) dl3(x, y, z) = l2(x, l2(y, z)) + l2(y, l2(z, x)) + l2(z, l2(x, y)),

(iii) l3(x, y, dm) = l2(x, l2(y,m)) + l2(y, l2(m,x)) + l2(m, l2(x, y)),

(iv) the Jacobiator identity:4∑i=1

(−1)i+1l2(xi, l3(x1, · · · , xi, · · · , x4))

+∑i<j

(−1)i+jl3(l2(xi, xj), x1, · · · , xi, · · · , xj, · · · , x4) = 0.

Usually, we denote a 2-term L∞-algebra by (g0, g1, d, l2, l3), or simply by G. A 2-termL∞-algebra is called strict if l3 = 0. Associated to a 2-term strict L∞-algebra, there is asemidirect product Lie algebra g0 n g1 = (g0⊕ g−1, [·, ·]s), where the bracket [·, ·]s is givenby

[x+m, y + n]s := l2(x, y) + l2(x, n) + l2(m, y). (2)


2.4. Definition. Let G = (g0, g1, d, l2, l3) and G ′ = (g′0, g′1, d′, l′2, l′3) be 2-term L∞-algebras. A homomorphism F from G to G ′ consists of: linear maps F0 : g0 → g′0, F1 :g1 → g′1 and F2 : g0 ∧ g0 → g′1, such that the following equalities hold for all x, y, z ∈g0, a ∈ g1,

(i) F0 ◦ d = d′ ◦ F1,

(ii) F0l2(x, y)− l′(F0(x), F0(y)) = d′F2(x, y),

(iii) F1l2(x, a)− l′(F0(x), F1(a)) = F2(x, da),

(iv) F2(l2(x, y), z)+c.p.+F1(l3(x, y, z)) = l′2(F0(x),F2(y, z))+c.p.+l′3(F0(x), F0(y), F0(z)).

It is well-known that the category of Lie 2-algebras and the category of 2-term L∞-algebras are equivalent. Thus, when we say “a Lie 2-algebra”, we mean a 2-term L∞-algebra in the sequel.

Let V : V1d−→ V0 be a complex of vector spaces. Define End0

d(V) by

End0d(V) , {(A0, A1) ∈ gl(V0)⊕ gl(V1)|A0 ◦ d = d ◦ A1},

and define End1(V) , Hom(V0, V1). There is a differential δ : End1(V) −→ End0d(V) given

byδ(φ) , φ ◦ d + d ◦ φ, ∀ φ ∈ End1(V),

and a bracket operation [·, ·] given by the graded commutator. More precisely, for anyA = (A0, A1), B = (B0, B1) ∈ End0

d(V) and φ ∈ End1(V), [·, ·] is given by

[A,B] = A ◦B −B ◦ A = (A0 ◦B0 −B0 ◦ A0, A1 ◦B1 −B1 ◦ A1),

and[A, φ] = A ◦ φ− φ ◦ A = A1 ◦ φ− φ ◦ A0. (3)

These two operations make End1(V) δ−→ End0d(V) into a strict Lie 2-algebra, which we

denote by End(V). It plays the same role as gl(V ) for a vector space V ([20]).A representation of a Lie 2-algebra G on V is a homomorphism (ρ0, ρ1, ρ2) from G to

End(V). A representation of a strict Lie 2-algebra G on V is called strict if ρ2 = 0. Givena strict representation of a strict Lie 2-algebra G on V , there is a semidirect product strictLie 2-algebra GnV , in which the degree 0 part is g0⊕V0, the degree 1 part is g1⊕V1, thedifferential is d+ d : g1⊕ V1 −→ g0⊕ V0, and for all x, y ∈ g0, a ∈ g1, u, v ∈ V0,m ∈ V1, l

s2

is given by

ls2(x+ u, y + v) = l2(x, y) + ρ0(x)v − ρ0(y)u,ls2(x+ u, a+m) = l2(x, a) + ρ0(x)m− ρ1(a)u.

2.5. Pre-Lie algebras and their representations.

274 YUNHE SHENG

2.6. Definition. A pre-Lie algebra (A, ·A) is a vector space A equipped with a bilinearproduct ·A : ⊗2A −→ A such that for any x, y, z ∈ A, the associator (x, y, z) = (x ·A y) ·Az − x ·A (y ·A z) is symmetric in x, y, i.e.,

(x, y, z) = (y, x, z), or equivalently, (x ·A y) ·A z−x ·A (y ·A z) = (y ·A x) ·A z−y ·A (x ·A z).

Let A be a pre-Lie algebra. The commutator [x, y]A = x ·A y − y ·A x defines a Liealgebra structure on A, which is called the sub-adjacent Lie algebra of A and denotedby g(A). Furthermore, L : A→ gl(A) with Lxy = x ·A y gives a representation of the Liealgebra g(A) on A. See [9] for more details.

2.7. Definition. Let (A, ·A) be a pre-Lie algebra and V a vector space. A representationof A on V consists of a pair (ρ, µ), where ρ : A −→ gl(V ) is a representation of the Liealgebra g(A) on V and µ : A −→ gl(V ) is a linear map satisfying

ρ(x)µ(y)u− µ(y)ρ(x)u = µ(x ·A y)u− µ(y)µ(x)u, ∀x, y ∈ A, u ∈ V. (4)

Usually, we denote a representation by (V ; ρ, µ). In this case, we will also say that(ρ, µ) is an action of (A, ·A) on V . Define R : A −→ gl(A) by Rxy = y ·A x. Then(A;L,R) is a representation of (A, ·A). Furthermore, (A∗; ad∗ = L∗ − R∗,−R∗) is also arepresentation of (A, ·A), where L∗ and R∗ are given by

〈L∗xξ, y〉 = 〈ξ,−Lxy〉, 〈R∗xξ, y〉 = 〈ξ,−Rxy〉, ∀x, y ∈ A, ξ ∈ A∗.

The cohomology complex for a pre-Lie algebra (A, ·A) with a representation (V ; ρ, µ)is given as follows ([15]). The set of (n+ 1)-cochains is given by

Cn+1(A, V ) = Hom(∧nA⊗ A, V ), n ≥ 0.

For all ω ∈ Cn(A, V ), the coboundary operator d : Cn(A, V ) −→ Cn+1(A, V ) is given by

dω(x1, x2, · · · , xn+1)

=n∑i=1

(−1)i+1ρ(xi)ω(x1, x2, · · · , xi, · · · , xn+1)

+n∑i=1

(−1)i+1µ(xn+1)ω(x1, x2, · · · , xi, · · · , xn, xi)

−n∑i=1

(−1)i+1ω(x1, x2, · · · , xi, · · · , xn, xi ·A xn+1)

+∑

1≤i<j≤n(−1)i+jω([xi, xj]A, x1, · · · , xi, · · · , xj, · · · , xn+1),

for all xi ∈ Γ(A), i = 1, 2 · · · , n+ 1.


2.8. O-operators and solutions of the classical Yang-Baxter equations.Let (g, [·, ·]g) be a Lie algebra and (V ; ρ) be a representation. A linear map T : V → g iscalled an O-operator on g associated to the representation (V ; ρ) if T satisfies

[T (u), T (v)]g = T(ρ(T (u))v − ρ(T (v))u

), ∀ u, v ∈ V. (5)

Associated to a representation (V ; ρ), we have the semidirect product Lie algebrag nρ∗ V ∗, where ρ∗ : g −→ gl(V ∗) is the dual representation. A linear map T : V −→ gcan be view as an element T ∈ ⊗2(g⊕ V ∗) via

T (ξ + u, η + v) = 〈T (u), η〉, ∀ ξ + u, η + v ∈ g∗ ⊕ V. (6)

Let σ be the exchanging operator acting on the tensor space, then r , T − σ(T ) isskew-symmetric.

2.9. Theorem. Let T : V → g be a linear map. Then r = T − σ(T ) is a solutionof the classical Yang-Baxter equation in the Lie algebra g nρ∗ V ∗ if and only if T is anO-operator.

2.10. Theorem. [5] Let A be a pre-Lie algebra. Then

r =n∑i=1

(ei ⊗ e∗i − e∗i ⊗ ei) (7)

is a solution of the CYBE in g(A)nL∗ A∗, where {ei} is a basis of A, and {e∗i } is the dualbasis.

3. 2-term pre-Lie∞-algebras and Pre-Lie 2-algebrasIn this section, we show that a 2-term pre-Lie∞-algebra can give rise to a Lie 2-algebra,and the left multiplication gives rise to a representation of the Lie 2-algebra. We introducethe notion of a pre-Lie 2-algebra, which is a categorification of a pre-Lie algebra. We provethat the category of 2-term pre-Lie∞-algebras and the category of pre-Lie 2-algebras areequivalent.

3.1. 2-term Pre-Lie∞-algebras. The notion of a pre-Lie∞-algebra was introducedin [11], which is a right-symmetric algebra1 up to homotopy. By a slight modification,we can obtain a left-symmetric algebra (the pre-Lie algebra we use in this paper) up tohomotopy. By truncation, we obtain a 2-term pre-Lie∞-algebra.

1A right-symmetric algebra (A, ·A) is a vector space A equipped with a bilinear product ·A : ⊗2A −→ Asuch that the associator satisfies (x, y, z) = (x, z, y), for all x, y, z ∈ A.

276 YUNHE SHENG

3.2. Definition. A 2-term pre-Lie∞-algebra is a 2-term graded vector space A = A0 ⊕A1, together with linear maps d : A1 −→ A0, · : Ai ⊗ Aj −→ Ai+j, 0 ≤ i + j ≤ 1, andl3 : ∧2A0 ⊗ A0 −→ A1, such that for all v, vi ∈ A0 and m,n ∈ A1, we have

(a1) d(v ·m) = v · dm,

(a2) d(m · v) = (dm) · v,

(a3) dm · n = m · dn,

(b1) v0 · (v1 · v2)− (v0 · v1) · v2 − v1 · (v0 · v2) + (v1 · v0) · v2 = dl3(v0, v1, v2),

(b2) v0 · (v1 ·m)− (v0 · v1) ·m− v1 · (v0 ·m) + (v1 · v0) ·m = l3(v0, v1, dm),

(b3) m · (v1 · v2)− (m · v1) · v2 − v1 · (m · v2) + (v1 ·m) · v2 = l3(dm, v1, v2),

(c)

v0 · l3(v1, v2, v3)− v1 · l3(v0, v2, v3) + v2 · l3(v0, v1, v3)+l3(v1, v2, v0) · v3 − l3(v0, v2, v1) · v3 + l3(v0, v1, v2) · v3

−l3(v1, v2, v0 · v3) + l3(v0, v2, v1 · v3)− l3(v0, v1, v2 · v3)−l3(v0 · v1 − v1 · v0, v2, v3) + l3(v0 · v2 − v2 · v0, v1, v3)− l3(v1 · v2 − v2 · v1, v0, v3)= 0.

Usually, we denote a 2-term pre-Lie∞-algebra by (A0, A1, d, ·, l3), or simply by A. A2-term pre-Lie∞-algebra (A0, A1, d, ·, l3) is said to be skeletal (strict) if d = 0 (l3 = 0).

Given a 2-term pre-Lie∞-algebra (A0, A1, d, ·, l3), we define l2 : Ai ∧ Aj −→ Ai+j andl3 : ∧3A0 −→ A1 by

l2(u, v) = u · v − v · u, (8)l2(u,m) = −l2(m,u) = u ·m−m · u, (9)

l3(u, v, w) = l3(u, v, w) + l3(v, w, u) + l3(w, u, v). (10)

Furthermore, define L0 : A0 −→ End(A0)⊕ End(A1) by

L0(u)v = u · v, L0(u)m = u ·m. (11)

Define L1 : A1 −→ Hom(A0, A1) by

L1(m)u = m · u. (12)

Define L2 : ∧2A0 −→ Hom(A0, A1) by

L2(u, v)w = −l3(u, v, w), ∀u, v, w ∈ A0. (13)


3.3. Theorem. Let A = (A0, A1, d, ·, l3) be a 2-term pre-Lie∞-algebra. Then, we have(A0, A1, d, l2, l3) is a Lie 2-algebra, which we denote by G(A), where l2 and l3 are givenby (8)-(10) respectively. Furthermore, (L0, L1, L2) is a representation of the Lie 2-algebraG(A) on the complex of vector spaces A1

d−→ A0, where L0, L1, L2 are given by (11)-(13)respectively.

Proof. By Conditions (a1)-(a3), we have

dl2(v,m) = d(v ·m−m · v) = v · dm− (dm) · v = l2(v, dm),l2(dm,n) = (dm) · n− n · dm = m · dn− (dn) ·m = l2(m, dn).

By Condition (b1), we have

l2(v0, l2(v1, v2)) + c.p. = l2(v0, v1 · v2 − v2 · v1) + c.p.

= v0 · (v1 · v2)− (v1 · v2) · v0 − v0 · (v2 · v1) + (v2 · v1) · v0 + c.p.

= d(l3(v0, v1, v2) + l3(v1, v2, v0) + l3(v2, v0, v1))= dl3(v0, v1, v2).

Similarly, by Conditions (b2) and (b3), we have

l2(v0, l2(v1,m)) + l2(v1, l2(m, v0)) + l2(m, l2(v0, v1))= l2(v0, v1 ·m−m · v1) + l2(v1,m · v0 − v0 ·m) + l2(m, v0 · v1 − v1 · v0)= v0 · (v1 ·m)− (v1 ·m) · v0 − v0 · (m · v1) + (m · v1) · v0

+v1 · (m · v0)− (m · v0) · v1 − v1 · (v0 ·m) + (v0 ·m) · v1

+m · (v0 · v1)− (v0 · v1) ·m−m · (v1 · v0) + (v1 · v0) ·m= l3(v0, v1, dm) + l3(v1, dm, v0) + l3(dm, v0, v1)= l3(v0, v1, dm).

At last, by Condition (c), we can get

l2(v0, l3(v1, v2, v3))− l2(v1, l3(v0, v2, v3)) + l2(v2, l3(v0, v1, v3))− l2(v3, l3(v0, v1, v2))= l3(l2(v0, v1), v2, v3)− l3(l2(v0, v2), v1, v3) + l3(l2(v0, v3), v1, v2) + l3(l2(v1, v2), v0, v3)−l3(l2(v1, v3), v0, v2) + l3(l2(v2, v3), v0, v1).

Thus, (A0, A1, d, l2, l3) is a Lie 2-algebra.By Condition (a1), we deduce that L0(u) ∈ End0

d(A) for all u ∈ A0. By Conditions(a2) and (a3), we have

δ ◦ L1(m) = L0(dm). (14)Furthermore, we have

L0(l2(u, v))w = (u · v) · w − (v · u) · w = u · (v · w)− v · (u · w)− dl3(u, v, w)= [L0(u), L0(v)]w − dl3(u, v, w),

278 YUNHE SHENG

which implies that

L0(l2(u, v))− [L0(u), L0(v)] = d ◦ L2(u, v). (15)

Similarly, we have

L1(l2(u,m))− [L0(u), L1(m)] = L2(u, dm). (16)

At last, by Condition (c) in Definition 3.2, we get

− [L0(u), L2(v, w)] + L2(l2(u, v), w) + c.p.+ L1(l3(u, v, w)) = 0. (17)

By (14)-(17), we deduce that (L0, L1, L2) is a homomorphism from the Lie 2-algebra G(A)to End(V). The proof is finished.

3.4. Example. Let V be a vector space. Denote by A0 = gl(V )⊕V and A1 = V . Defined : A1 −→ A0, · : Ai ⊗ Aj −→ Ai+j, 0 ≤ i+ j ≤ 1, and l3 : ∧2A0 ⊗ A0 −→ A1 by

du = u, ∀ u ∈ A1,(A+ u) · (B + v) = AB + 1

2Av, ∀ A+ u,B + v ∈ A0,(A+ u) · v = 1

2Av, ∀ A+ u ∈ A0, v ∈ A1,u · (B + v) = 0, ∀ u ∈ A1, B + v ∈ A0,

l3(A+ u,B + v, C + w) = −14 [A,B]w, ∀ A+ u,B + v, C + w ∈ A0.

Then it is straightforward to prove that A = (A0, A1, d, ·, l3) is a 2-term pre-Lie∞-algebra.By Theorem 3.3, we obtain a Lie 2-algebra G(A). Actually, this Lie 2-algebra is the

Lie 2-algebra associated to Weinstein’s omni-Lie algebra. See [25] for more details.

3.5. Definition. Let A = (A0, A1, d, ·, l3) and A′ = (A′0, A′1, d′, ·′, l′3) be 2-term pre-Lie∞-algebras. A homomorphism (F0, F1, F2) from A to A′ consists of linear maps F0 : A0 −→A′0, F1 : A1 −→ A′1, and F2 : A0 ⊗ A0 −→ A′1 such that the following equalities hold:

(i) F0 ◦ d = d′ ◦ F1,

(ii) F0(u · v)− F0(u) ·′ F0(v) = d′F2(u, v),

(iii) F1(u ·m)− F0(u) ·′ F1(m) = F2(u, dm), F1(m · u)− F1(m) ·′ F0(u) = F2(dm,u),

(iv) F0(u) ·′ F2(v, w)−F0(v) ·′ F2(u,w) +F2(v, u) ·′ F0(w)−F2(u, v) ·′ F0(w)−F2(v, u ·w)+F2(u, v ·w)−F2(u · v, w) +F2(v ·u,w) + l′3(F0(u), F0(v), F0(v))−F1l3(u, v, w) = 0.

By straightforward computations, we have


3.6. Proposition. Let A = (A0, A1, d, ·, l3) and A′ = (A′0, A′1, d′, ·′, l′3) be 2-term pre-Lie∞-algebras, (F0, F1, F2) a homomorphism from A to A′. Then (F0, F1,F2) is a homo-morphism from the corresponding Lie 2-algebra G(A) to G(A′), where F2 : ∧2A0 −→ A1is given by

F2(u, v) = F2(u, v)− F2(v, u), ∀u, v ∈ A0. (18)

At the end of this subsection, we introduce composition and identity for 2-term pre-Lie∞-algebra homomorphisms. Let F = (F0, F1, F2) : A −→ A′ and G = (G0, G1, G2) :A′ −→ A′′ be 2-term pre-Lie∞-algebra homomorphisms. Their composition GF =((GF )0, (GF )1, (GF )2) is defined by (GF )0 = G0 ◦ F0, (GF )1 = G1 ◦ F1, and (GF )2is given by

(GF )2(u, v) = G2(F0(u), F0(v)) +G1(F2(u, v)). (19)

It is straightforward to verify that GF = ((GF )0, (GF )1, (GF )2) : A −→ A′′ is a 2-term pre-Lie∞-algebra homomorphism. It is obvious that (idA0 , idA1 , 0) is the identityhomomorphism. Thus, we obtain

3.7. Proposition. There is a category, which we denote by 2preLie, with 2-term pre-Lie∞-algebras as objects, homomorphisms between them as morphisms.

3.8. Pre-Lie 2-algebras.

3.9. Definition. A pre-Lie 2-algebra is a 2-vector space V endowed with a bilinear func-tor ? : V× V −→ V and a natural isomorphism Ju,v,w for all u, v, w ∈ V0,

Ju,v,w : (u ? v) ? w − u ? (v ? w) −→ (v ? u) ? w − v ? (u ? w), (20)

such that the following identity is satisfied:

(0J1,2,3) ·v (−J0,21,3 + 1(0(21))3 + J0,2,13 − 1(02)(13))·v(1(0(21))3−((21)0)3+(21)(03) + J02,1,3 − 1((02)1)3 − J20,1,3 + 1((20)1)3 + 2J0,1,3 − 21(01)3)

= (−J0,12,3 + 1(0(12))3 + J0,1,23 − 1(01)(23))·v(J1,2,03 + 11(2(03)) + J01,2,3 − 1((01)2)3 − J10,2,3 + 1((10)2)3 + 1J0,2,3 − 11(02)3 + 1−((12)0)3+(0(12))3)·v(−J1,2,03− 11(20)+2(10)3− J2,0,13 + 1(20)13− J0,1,23 + 1(10)23+1(21)(03)−2(1(03))−2((01)3)+2((10)3)−1((02)3)+1((20)3)). (21)

Here, 0, 1, 2, 3 denote v0, v1, v2, v3 respectively, ij denotes vi?vj, iJj,k,l denotes vi?Jvj ,vk,vl,

and Jj,k,li denotes Jvj ,vk,vl? vi. Or, in terms of a commutative diagram,

0((12)3− 1(23))

−J0,12,3+1(0(12))3+J0,1,23−1(01)(23)

��

0J1,2,3 // 0((21)3− 2(13))

−J0,21,3+1(0(21))3+J0,2,13−1(02)(13)

��P

ε

��

Q

ε

��M

κ // N

280 YUNHE SHENG

where

P = −((12)0)3 + (12)(03) + (0(12))3− (01)(23) + (10)(23)− 1(0(23)),Q = (0(21))3− ((21)0)3 + (21)(03)− (02)(13) + (20)(13)− 2(0(13)),ε = J1,2,03 + 11(2(03)) + J01,2,3 − 1((01)2)3 − J10,2,3 + 1((10)2)3 + 1J0,2,3 − 11(02)3 + 1−((12)0)3+(0(12))3,

ε = 1(0(21))3−((21)0)3+(21)(03) + J02,1,3 − 1((02)1)3 − J20,1,3 + 1((20)1)3 + 2J0,1,3 − 21(01)3,

κ = −J1,2,03− 11(20)+2(10)3− J2,0,13 + 1(20)13− J0,1,23 + 1(10)23+1(21)(03)−2(1(03))−2((01)3)+2((10)3)−1((02)3)+1((20)3),

M = −((12)0)3 + (0(12))3 + 1(2(03)) + (21)(03)− 2(1(03))− ((01)2)3 + (2(01))3− 2((01)3)+((10)2)3− (2(10))3 + 2((10)3)− 1((02)3) + 1((20)3)− 1(2(03)),

N = (0(21))3− ((21)0)3 + (21)(03)− ((02)1)3 + (1(02))3− 1((02)3)+((20)1)3− (1(20))3 + 1((20)3)− 2((01)3) + 2((10)3)− 2(1(03)).

3.10. Definition. Let (V, ?, J) and (V′, ?′, J ′) be pre-Lie 2-algebras. A homomorphismΦ : V −→ V′ consists of

• A linear functor (Φ0,Φ1) from V to V′,

• A bilinear natural transformation Φ2 : Φ0(u) ?′ Φ0(v) −→ Φ0(u ? v),

such that the following identity holds:

J ′Φ0(u),Φ0(v),Φ0(w) ·v (F2(v, u) ? 1Φ0(w) − 1Φ0(v) ? F2(u,w)) ·v (F2(v ? u, w)− F2(v, u ? w))= (F2(u, v) ? 1Φ0(w) − 1Φ0(u) ? F2(v, w)) ·v (F2(u ? v, w)− F2(u, v ? w)) ·v F1Ju,v,w,

or, in terms of a commutative diagram:

(Φ0(u) ?′ Φ0(v)) ?′ Φ0(w)− Φ0(u) ?′ (Φ0(v) ?′ Φ0(w))

F2(u,v)?1Φ0(w)−1Φ0(u)?F2(v,w)��

J′Φ0(u),Φ0(v),Φ0(w)// (Φ0(v) ?′ Φ0(u)) ?′ Φ0(w)− Φ0(v) ?′ (Φ0(u) ?′ Φ0(w))

F2(v,u)?1Φ0(w)−1Φ0(v)?F2(u,w)��

Φ0(u ? v) ? Φ0(w)− Φ0(u) ? Φ0(v ? w)

F2(u?v,w)−F2(u,v?w)��

Φ0(v ? u) ? Φ0(w)− Φ0(v) ? Φ0(u ? w)

F2(v?u,w)−F2(v,u?w)��

F0((u ? v) ? w)− F0(u ? (v ? w))F1Ju,v,w // F0((v ? u) ? w)− F0(v ? (u ? w)).

The composition of two homomorphisms Φ : V −→ V′ and Ψ : V′ −→ V′′, which wedenote by ΨΦ : V −→ V′′ is defined as follows:

(ΨΦ)0 = Ψ0 ◦ Φ0, (ΨΦ)1 = Ψ1 ◦ Φ1, (ΨΦ)2(u, v) = Ψ2(Φ0(u),Φ0(v)) ·v Ψ1(Φ2(u, v)).

The identity homomorphism 1V has the identity functor as its underlying functor, togetherwith an identity natural transformation. It is straightforward to obtain

3.11. Proposition. There is a category, which we denote by preLie2, with pre-Lie2-algebras as objects, homomorphisms between them as morphisms.


3.12. The equivalence.

3.13. Theorem. The categories 2preLie and preLie2, which are given in Proposition3.7 and Proposition 3.11 respectively, are equivalent.

Thus, in the following sections, a 2-term pre-Lie∞-algebra will be called a pre-Lie2-algebra.

Proof. We only give a sketch of the proof. First we construct a functor T : 2preLie −→preLie2.

Given a 2-term pre-Lie∞-algebra A = (A0, A1, d, ·, l3), we have a 2-vector space Agiven by (1). More precisely, we have A0 = A0, A1 = A0 ⊕ A1. Define a bilinear functor? : A× A −→ A by

(u+m) ? (v + n) = u · v + u · n+m · v + dm · n, ∀ u+m, v + n ∈ A1 = A0 ⊕ A1.

Define the Jacobiator J : ⊗3A0 −→ A1 by

Ju,v,w = (u · v) · w − u · (v · w) + l3(x, y, z).

By the various conditions of A being a 2-term pre-Lie∞-algebra, we deduce that (A, ?, J)is a pre-Lie 2-algebra. Thus, we have constructed a pre-Lie 2-algebra A = T (A) from a2-term pre-Lie∞-algebra A.

For any homomorphism F = (F0, F1, F2) form A to A′, next we construct a pre-Lie2-algebra homomorphism Φ = T (F ) from A = T (A) to A′ = T (A′). Let Φ0 = F0, Φ1 =F0 ⊕ F1, and Φ2 be given by

Φ2(u, v) = F0(u) ·′ F0(v) + F2(u, v).

Then Φ2(u, v) is a natural isomorphism from Φ0(u)·′Φ0(v) to Φ0(u·v), and Φ = (Φ0,Φ1,Φ2)is a homomorphism from A to A′.

One can also deduce that T preserves the identity homomorphisms and the compo-sition of homomorphisms. Thus, T constructed above is a functor from 2pre-Lie topreLie2.

Conversely, given a pre-Lie 2-algebra A, we construct the 2-term pre-Lie∞-algebraA = S(A) as follows. As a complex of vector spaces, A is obtained as follows: A0 =A0, A1 = Ker(s), and d = t|Ker(s), where s, t are the sauce map and the target map in the2-vector space A. Define a multiplication · : Ai ⊗ Aj −→ Ai+j, 0 ≤ i+ j ≤ 1, by

u · v = u ? v, u ·m = 1u ? m, m · u = m ? 1u, ∀u, v ∈ A0, m, n ∈ A1.

Define l3 : ∧2A0 ⊗ A0 −→ A1 by

l3(u, v, w) = Ju,v,w − 1s(Ju,v,w).

The various conditions of A being a pre-Lie 2-algebra imply that A is 2-term pre-Lie∞-algebra.

282 YUNHE SHENG

Let Φ = (Φ0,Φ1,Φ2) : A −→ A′ be a pre-Lie 2-algebra homomorphism, and S(A) =A, S(A′) = A′. Define S(Φ) = F = (F0, F1, F2) as follows. Let F0 = Φ0, F1 = Φ1|A1=Ker(s)and define F2 by

F2(u, v) = Φ2(u, v)− 1s(Φ2(u,v)).

It is not hard to deduce that F is a homomorphism between 2-term pre-Lie∞-algebras.Furthermore, S also preserves the identity homomorphisms and the composition of ho-momorphisms. Thus, S is a functor from preLie2 to 2pre-Lie.

We are left to show that there are natural isomorphisms α : T ◦ S =⇒ idpreLie2 andβ : S ◦ T =⇒ id2preLie. For a pre-Lie 2-algebra (A, ?, J), applying the functor S to A, weobtain a 2-term pre-Lie∞-algebra A = (A0, A1, d = t|Ker(s), ·, l3), where A0 = A0, A1 =Ker(s). Applying the functor T to A, we obtain a pre-Lie 2-algebra (A′, ?′, J ′), with thespace A0 of objects and the space A0 ⊕ Ker(s) of morphisms. Define αA : A′ −→ A bysetting

(αA)0(u) = u, (αA)1(u+m) = 1u +m.

It is obvious that αA is an isomorphism of 2-vector spaces. Furthermore, since ? is abilinear functor, we have 1u ? 1v = 1u?v, and

m ? n = (m ·v 1dm) ? (10 ·v n) = (m ? 10) ·v (1dm ? n) = 1dm ? n.

Therefore, we have

αA((u+m) ?′ (v + n)) = αA(u · v + u · n+m · v + dm · n)= αA(u ? v + 1u ? n+m ? 1v + 1dm ? n)= 1u?v + 1u ? n+m ? 1v + 1dm ? n

= 1u ? 1v + 1u ? n+m ? 1v + 1dm ? n

= αA(u+m) ? αA(v + n),

which implies that αA is also a pre-Lie 2-algebra homomorphism with (αA)2 the identityisomorphism. Thus, αA is an isomorphism of pre-Lie 2-algebras. It is also easy to see thatit is a natural isomorphism.

For a 2-term pre-Lie∞-algebra A = (A0, A1, d, ·, l3), applying the functor S to A, weobtain a pre-Lie 2-algebra (A, ?, J). Applying the functor T to A, we obtain exactlythe same 2-term pre-Lie∞-algebra A. Thus, βA = idA = (idA0 , idA1) is the naturalisomorphism from T ◦ S to id2preLie. This finishes the proof.

3.14. Remark. We can further obtain 2-categories 2preLie and preLie2 by introducing2-morphisms and strengthen Theorem 3.13 to the 2-equivalence of 2-categories. We omitdetails.

4. Skeletal and strict pre-Lie 2-algebrasIn this section, we study skeletal pre-Lie 2-algebras and strict pre-Lie 2-algebras in detail.


Let (A0, A1, d = 0, ·, l3) be a skeletal pre-Lie 2-algebra. Condition (b1) in Definition3.2 implies that (A0, ·) is a pre-Lie algebra. Define ρ and µ from A0 to gl(A1) by

ρ(u)m = u ·m, µ(u)m = m · u, ∀ u ∈ A0, m ∈ A1. (22)

Condition (b2) and (b3) in Definition 3.2 implies that (A1; ρ, µ) is a representation of thepre-Lie algebra (A0, ·). Furthermore, Condition (c) exactly means that l3 is a 3-cocycleon A0 with values in A1. Summarize the discussion above, we have

4.1. Theorem. There is a one-to-one correspondence between skeletal pre-Lie 2-algebrasand triples ((A0, ·), (A1; ρ, µ), l3), where (A0, ·) is a pre-Lie algebra, (A1; ρ, µ) is a repre-sentation of (A0, ·), and l3 is a 3-cocycle on (A0, ·) with values in A1.

Recall that a skew-symmetric bilinear form ω : ∧2A −→ A on a pre-Lie algebra (A, ·A)is called invariant if

ω(u ·A v − v ·A u,w) + ω(v, u ·A w) = 0, ∀u, v, w ∈ A. (23)

Equivalently, ω([u, v]A, w) + ω(v, u ·A w) = 0, where [·, ·]A is the Lie bracket in the sub-adjacent Lie algebra of A.

4.2. Lemma. Let ω be a skew-symmetric invariant bilinear form on a pre-Lie algebra(A, ·A). Then we have

ω(u ·A v, w) = ω(u,w ·A v). (24)

Proof. By (23), we have

ω(u ·A w − w ·A u, v) + ω(w, u ·A v) = 0. (25)

Since ω is skew-symmetric, by (23) and (25), we have

−ω(w ·A u, v)− ω(v ·A u,w) = 0,

which implies that ω(u ·A v, w) = ω(u,w ·A v).

Define ϕ : ∧2A⊗ A −→ R by

ϕ(u, v, w) = ω(u ·A v − v ·A u,w). (26)

4.3. Proposition. Let ω be a skew-symmetric invariant bilinear form on a pre-Lie al-gebra (A, ·A). Then ϕ defined by (26) is a 3-cocycle on A with values in R, i.e. dϕ = 0.

284 YUNHE SHENG

Proof. For any u, v, w, p ∈ A, by (23) and (24), we have

dϕ(u, v, w, p) = −ϕ(v, w, u ·A p) + ϕ(u,w, v ·A p)− ϕ(u, v, w ·A p)−ϕ([u, v]A, w, p) + ϕ([u,w]A, v, p)− ϕ([v, w]A, u, p)

= −ω([v, w]A, u ·A p) + ω([u,w]A, v ·A p)− ω([u, v]A, w ·A p)−ω([[u, v]A, w]A + c.p., p)

= ω(w, v ·A (u ·A p))− ω(w, u ·A (v ·A p)) + ω(w, [u, v]A ·A p)= ω(w, v ·A (u ·A p)− u ·A (v ·A p) + (u ·A v) ·A p− (v ·A u) ·A p)= 0,

which finishes the proof.

4.4. Example. Let ω be a skew-symmetric invariant bilinear form on a pre-Lie algebra(A, ·A). Consider the graded vector space A = A0 ⊕ A1 where A0 = A,A1 = R. Defined : R −→ A, · : Ai ⊗ Aj −→ Ai+j, 0 ≤ i+ j ≤ 1, and l3 : ⊗A0 −→ A1 by

d = 0,u · v = u ·A v,u ·m = m · u = 0,

l3(u, v, w) = ϕ(u, v, w),

for any u, v, w ∈ A and m ∈ A1. By Proposition 4.3, it is straightforward to verify thatA = (A,R, d = 0, ·, l3 = ϕ) is a pre-Lie 2-algebra. Furthermore, ω ia a closed 2-form onthe Lie algebra g(A), i.e.

ω([u, v]A, w) + ω([v, w]A, u) + ω([w, u]A, v) = 0,

which implies that l3(u, v, w)+ l3(v, w, u)+ l3(w, u, v) = 0. Thus, the skeletal Lie 2-algebraG(A) is strict.

Now we turn to the study on strict pre-Lie 2-algebras. First we introduce the notionof crossed modules of pre-Lie algebras, which can give rise to crossed modules of Liealgebras.

4.5. Definition. A crossed module of pre-Lie algebras is defined to be a quadruple((A0, ·0), (A1, ·1), d, (ρ, µ)) where (A0, ·0) and (A1, ·1) are pre-Lie algebras, d : A1 −→ A0is a homomorphism of pre-Lie algebras, and (ρ, µ) is an action of (A0, ·0) on A1 such thatfor all u ∈ A0 and m,n ∈ A1, the following equalities are satisfied:

(C1) d(ρ(u)m) = u ·0 dm, d(µ(u)m) = (dm) ·0 u,

(C2) ρ(dm)n = µ(dn)m = m ·1 n.

4.6. Example. Let (A, ·A) be a pre-Lie algebra and B ⊂ A an ideal. Then it is straight-forward to see that ((A, ·), (B, ·|B), i, (ρ, µ)) is a crossed module of pre-Lie algebras, wherei is the inclusion, and (ρ, µ) are given by ρ(u)v = u·Av, µ(u)v = v·Au, for all u ∈ A, v ∈ B.


4.7. Proposition. Let ((A0, ·0), (A1, ·1), d, (ρ, µ)) be a crossed module of pre-Lie alge-bras. Then we have

ρ(u)(m ·1 n) = (ρ(u)m) ·1 n+m ·1 ρ(u)n− (µ(u)m) ·1 n, (27)µ(u)(m ·1 n) = µ(u)(n ·1 m) +m ·1 µ(u)n− n ·1 µ(u)m. (28)

Consequently, there is a pre-Lie algebra structure · on the direct sum A0 ⊕ A1 given by

(u+m) · (v + n) = u ·0 v + ρ(u)n+ µ(v)m+m ·1 n. (29)

Proof. Since (ρ, µ) is an action of A0 on A1, we have

ρ(u)ρ(dm)n = ρ(u ·0 dm)n− ρ(dm ·0 u)n+ ρ(dm)ρ(u)n= ρ(dρ(u)m)n− ρ(dµ(u)m)n+ ρ(dm)ρ(u)n.

The second equality is due to (C1). By (C2), we obtain (27). (28) can be obtainedsimilarly. The other conclusion is obvious.

4.8. Theorem. There is a one-to-one correspondence between strict pre-Lie 2-algebrasand crossed modules of pre-Lie algebras.Proof. Let (A0, A1, d, ·, l3 = 0) be a strict pre-Lie 2-algebra. We construct a crossedmodule of pre-Lie algebras as follows. Obviously, (A0, ·) is a pre-Lie algebra. Define amultiplication ·1 on A1 by

m ·1 n = (dm) · n = m · dn. (30)

Then by Conditions (a1) and (b2) in Definition 3.2, we have

m ·1 (n ·1 p)− (m ·1 n) ·1 p− n ·1 (m ·1 p) + (n ·1 m) ·1 p= (dm) · ((dn) · p)− d((dm) · n) · p− (dn) · ((dm) · p) + d((dn) ·m) ·1 p= (dm) · ((dn) · p)− ((dm) · dn) · p− (dn) · ((dm) · p) + ((dn) · dm) ·1 p = 0,

which implies that (A1, ·1) is a pre-Lie algebra. Also by Condition (a1), we deduce that dis a homomorphism between pre-Lie algebras. Define ρ, µ : A0 −→ gl(A1) by

ρ(u)m = u ·m, µ(u)m = m · u. (31)

By Conditions (b2) and (b3) in Definition 3.2, it is straightforward to deduce that (ρ, µ)is an action of (A0, ·) on A1. By Conditions (a1) and (a2), we deduce that Condition(C1) hold. Condition (C2) follows from the definition of ·1 directly. Thus, the data((A0, ·), (A1, ·1), d, (ρ, µ)) constructed above is a crossed module of pre-Lie algebras.

Conversely, a crossed module of pre-Lie algebras ((A0, ·), (A1, ·1), d, (ρ, µ)) gives rise toa strict pre-Lie 2-algebra (A0, A1, d, ·, l3 = 0), where · : Ai ⊗ Aj −→ Ai+j, 0 ≤ i + j ≤ 1is given by

u · v = u ·0 v, u ·m = ρ(u)m, m · u = µ(u)m.The crossed module conditions give various conditions for a strict pre-Lie 2-algebra. Weomit details.

286 YUNHE SHENG

A pre-Lie algebra has its sub-adjacent Lie algebra. Similarly, a crossed module of pre-Lie algebras has its sub-adjacent crossed module of Lie algebras. Recall that a crossedmodule of Lie algebras is a quadruple (h1, h0, dt, φ), where h1 and h0 are Lie algebras,dt : h1 −→ h0 is a Lie algebra homomorphism and φ : h0 −→ Der(h1) is an action of Liealgebra h0 on Lie algebra h1 as a derivation, such that

dt(φX(A)) = [X, dt(A)]h0 , φdt(A)(B) = [A,B]h1 , ∀X ∈ h0, A,B ∈ h1.

4.9. Proposition. Let ((A0, ·0), (A1, ·1), d, (ρ, µ)) be a crossed module of pre-Lie algebrasand g(A0), g(A1) the corresponding sub-adjacent Lie algebras of (A0, ·0), (A1, ·1) respec-tively. Then (g(A0), g(A1), d, ρ− µ) is a crossed module of Lie algebras.

Proof. The fact that d is a homomorphism between pre-Lie algebras implies that d isalso a homomorphism between Lie algebras. Since (A1; ρ, µ) is a representation of (A0, ·0),(A1; ρ−µ) is a representation of the Lie algebra g(A0). By (C1), we have d((ρ−µ)(u)m) =[u, dm]0. By (C2), we have (ρ − µ)(dm)n = [m,n]1. Thus, (g(A0), g(A1), d, ρ − µ) is acrossed module of Lie algebras.

5. Categorification of O-operatorsLet G = (g0, g1, d, l2, l3) be a Lie 2-algebra and (ρ0, ρ1, ρ2) be a representation of G on a2-term complex of vector spaces V = V1

d−→ V0.

5.1. Definition. A triple (T0, T1, T2), where T0 : V0 −→ g0, T1 : V1 −→ g1 is a chainmap, and T2 : ∧2V0 −→ g1 is a linear map, is called an O-operator on G associated tothe representation (ρ0, ρ1, ρ2), if for all u, v, vi ∈ V0 and m ∈ V1 the following conditionsare satisfied:

(i) T0(ρ0(T0u)v − ρ0(T0v)u

)− l2(T0u, T0v) = dT2(u, v);

(ii) T1(ρ1(T1m)v − ρ0(T0v)m

)− l2(T1m,T0v) = T2(dm, v);

(iii)

l2(T0(v1), T2(v2, v3)) + T2(v3, ρ0(T0v1)v2 − ρ0(T0v2)v1

)+T1

(ρ1(T2(v2, v3))v1 + ρ2(T0v2, T0v3)v1

)+ c.p.+ l3(T0v1, T0v2, T0v3) = 0.

5.2. Example. Let A = (A0, A1, d, ·, l3) be a pre-Lie 2-algebra. Then, (T0 = idA0 , T1 =idA1 , T2 = 0) is an O-operator on the Lie 2-algebra G(A) associated to the representation(L0, L1, L2) given in Theorem 3.3.

Define a degree 0 multiplication · : Vi ⊗ Vj −→ Vi+j, 0 ≤ i+ j ≤ 1, on V by

u · v = ρ0(T0u)v, u ·m = ρ0(T0u)m, m · u = ρ1(T1m)u. (32)


Define l3 : ∧2V0 ⊗ V0 −→ V1 by

l3(v1, v2, v3) = −ρ1(T2(v1, v2))v3 − ρ2(T0v1, T0v2)v3. (33)

Now, Condition (iii) in Definition 5.1 reads

l2(T0(v1), T2(v2, v3)) + T2(v3, v1 · v2 − v2 · v1)− T1(l3(v1, v2, v3)) + c.p.

+l3(T0v1, T0v2, T0v3) = 0. (34)

5.3. Theorem. Let (ρ0, ρ1, ρ2) be a representation of G on V and (T0, T1, T2) an O-operator on G associated to the representation (ρ0, ρ1, ρ2). Then, (V0, V1, d, ·, l3) is a pre-Lie 2-algebra, where the multiplication “·” and l3 are given by (32) and (33) respectively.Proof. By the fact that d ◦ ρ(x) = ρ(x) ◦ d for all x ∈ g0, we deduce that

d(u ·m) = dρ0(T0u)m = ρ0(T0u)dm = u · dm.

By the fact that both (T0, T1) and (ρ0, ρ1) are chain maps, we have

d(m · u) = d(ρ1(T1m)u) = δ(ρ1(T1m))u = ρ0(dT1m)u = ρ0(T0dm)u = (dm) · u.

Similarly, we have

(dm) · n = ρ0(T0dm)n = ρ0(dT1m)n = δ(ρ1(T1m))n = ρ1(T1m)(dn) = m · (dn).

Thus, Conditions (a1)-(a3) in Definition 3.2 hold. For all u, v, w ∈ A0, we have

u · (v · w)− (u · v) · w − v · (u · w) + (v · u) · w= ρ0(T0u)ρ0(T0v)w − ρ0(T0(ρ0(T0u)v))w − ρ0(T0v)ρ0(T0u)w + ρ0(T0(ρ0(T0v)u))w

= [ρ0(T0u), ρ0(T0v)]w − ρ0

(T0(ρ0(T0u)v)− T0(ρ0(T0v)u)

)w

= ρ0(l2(T0u, T0v))w − dρ2(T0u, T0v)w − ρ0

(T0(ρ0(T0u)v)− T0(ρ0(T0v)u)

)w

= −ρ0(dT2(u, v))w − dρ2(T0u, T0v)w= −dρ1(T2(u, v))w − dρ2(T0u, T0v)w= dl3(u, v, w),

which implies that Condition (b1) in Definition 3.2 holds. Similarly, Conditions (b2) and(b3) also hold.

288 YUNHE SHENG

The left hand side of Condition (c) is equal to

ρ0(T0v0)l3(v1, v2, v3)− ρ0(T0v1)l3(v0, v2, v3) + ρ0(T0v2)l3(v0, v1, v3)+ρ1(T1l3(v1, v2, v0))v3 − ρ1(T1l3(v0, v2, v1))v3 + ρ1(T1l3(v0, v1, v2))v3

+ρ1(T2(v1, v2))(v0 · v3) + ρ2(T0v1, T0v2)(v0 · v3)− ρ1(T2(v0, v2))(v1 · v3)−ρ2(T0v0, T0v2)(v1 · v3) + ρ1(T2(v0, v1))(v2 · v3) + ρ2(T0v0, T0v1)(v2 · v3)+ρ1(T2(ρ0(T0v0)v1 − ρ0(T0v1)v0, v2))v3 + ρ2(T0(ρ0(T0v0)v1 − ρ0(T0v1)v0), T0v2)v3

−ρ1(T2(ρ0(T0v0)v2 − ρ0(T0v2)v0, v1))v3 − ρ2(T0(ρ0(T0v0)v2 − ρ0(T0v2)v0), T0v1)v3

+ρ1(T2(ρ0(T0v1)v2 − ρ0(T0v2)v1, v0))v3 + ρ2(T0(ρ0(T0v1)v2 − ρ0(T0v2)v1), T0v0)v3

= −ρ0(T0v0)ρ1(T2(v1, v2))v3 − ρ0(T0v0)ρ2(T0v1, T0v2)v3

+ρ0(T0v1)ρ1(T2(v0, v2))v3 + ρ0(T0v1)ρ2(T0v0, T0v2)v3

−ρ0(T0v2)ρ1(T2(v0, v1))v3 − ρ0(T0v2)ρ2(T0v0, T0v1)v3

+ρ1(T1l3(v1, v2, v0))v3 − ρ1(T1l3(v0, v2, v1))v3 + ρ1(T1l3(v0, v1, v2))v3

+ρ1(T2(v1, v2))ρ0(v0)v3 + ρ2(T0v1, T0v2)ρ0(v0)v3 − ρ1(T2(v0, v2))ρ0(v1)v3

−ρ2(T0v0, T0v2)ρ0(v1)v3 + ρ1(T2(v0, v1))ρ0(v2)v3 + ρ2(T0v0, T0v1)ρ0(v2)v3


−ρ1(T2(ρ0(T0v0)v2 − ρ0(T0v2)v0, v1))v3 − ρ2(T0(ρ0(T0v0)v2 − ρ0(T0v2)v0), T0v1)v3


=(− [ρ0(T0v0), ρ1(T2(v1, v2))] + c.p.

)v3 +

(− [ρ0(T0v0), ρ2(T0v1, T0v2)] + c.p.

)v3

+(ρ1(T1l3(v0, v1, v2)) + c.p.

)v3 +

(ρ1T2(v0 · v1 − v1 · v0, v2) + c.p.

)v3

+(ρ2(T0(v0 · v1 − v1 · v0), T0v2) + c.p.

)v3

=(− ρ1l2(T0v0, T2(v1, v2)) + ρ2(T0v0, dT2(v1, v2)) + c.p.

)v3

+(− [ρ0(T0v0), ρ2(T0v1, T0v2)] + c.p.

)v3 +

(ρ1(T1l3(v0, v1, v2)) + c.p.

)v3

+(ρ1T2(v0 · v1 − v1 · v0, v2) + c.p.

)v3 +

(ρ2(T0(v0 · v1 − v1 · v0), T0v2) + c.p.

)v3.

By Condition (ii) in Definition 5.1, we have

ρ2(T0v0, dT2(v1, v2))+c.p.+ρ2(T0(v0·v1−v1·v0), T0v2)+c.p. = ρ2(l2(T0v0, T0v1), T0v2)+c.p..

By the fact that (ρ0, ρ1, ρ2) is a representation, we have

[ρ0(T0v1), ρ2(T0v2, T0v3)] + c.p.− ρ2(l2(T0v1, T0v2), T0v3) + c.p. = ρ1l3(T0v1, T0v2, T0v3).

By (34), we deduce that Condition (c) in Definition 3.2 holds. Thus, (V0, V1, d, ·, l3) is apre-Lie 2-algebra. This finishes the proof.


5.4. Corollary. Let (ρ0, ρ1, ρ2) be a representation of the Lie 2-algebra G on V and(T0, T1, T2) be an O-operator on G associated to the representation (ρ0, ρ1, ρ2). Then(T0, T1, T2) is a homomorphism from the Lie 2-algebra G(V) to G.

6. Solutions of 2-graded Classical Yang-Baxter EquationsLet G = (g0, g1, d, l2) be a strict Lie 2-algebra and r ∈ g0 ⊗ g1 ⊕ g1 ⊗ g0 and r ∈ g1 ⊗ g1.Denote by R = r − (d⊗ 1 + 1⊗ d)r.

6.1. Definition. ([8]) The classical Yang-Baxter equation for R in the semidirect productLie algebra g0 n g1 = (g0 ⊕ g1, [·, ·]s) together with (d⊗ 1− 1⊗ d)R = 0 are called the 2-graded classical Yang-Baxter Equations (2-graded CYBE) in the strict Lie 2-algebraG, where [·, ·]s is the semidirect product Lie algebra structure given by (2).

More precisely, the 2-graded CYBE reads:(a) R is skew-symmetric,

(b) [R12, R13]s + [R13, R23]s + [R12, R23]s = 0,

(c) (d⊗ 1− 1⊗ d)r = 0.For R = ∑

iai ⊗ bi,

R12 =∑i

ai ⊗ bi ⊗ 1; R13 =∑i

ai ⊗ 1⊗ bi; R23 =∑i

1⊗ ai ⊗ bi. (35)

Let (ρ0, ρ1) be a strict representation of the Lie 2-algebra G = (g0, g1, d, l2) on the 2-termcomplex of vector space V : V1

d−→ V0. We view ρ0 ⊕ ρ1 a linear map from g0 ⊕ g1 togl(V0 ⊕ V1) by

(ρ0 ⊕ ρ1)(x+ a)(u+m) = ρ0(x)(u) + ρ0(x)m+ ρ1(a)u. (36)

By straightforward computations, we have

6.2. Lemma. With the above notations, ρ0⊕ρ1 : g0⊕g1 −→ gl(V0⊕V1) is a representationof (g0 ⊕ g1, [·, ·]s) on V0 ⊕ V1. Furthermore, (T0, T1) is an O-operator on G associated tothe representation (ρ0, ρ1) if and only if

(a) T0 + T1 : V0 ⊕ V1 −→ g0 ⊕ g1 is an O-operator on the Lie algebra (g0 ⊕ g1, [·, ·]s)associated to the representation ρ0 ⊕ ρ1,

(b) T0 ◦ d = d ◦ T1.

Let (ρ∗0, ρ∗1) be the dual representation of (ρ0, ρ1). Then we have the semidirect productLie 2-algebra G = G n(ρ∗

0,ρ∗1) V∗, where G0 = g0 ⊕ V ∗1 , G1 = g1 ⊕ V ∗0 , and d = d⊕ d∗. It is

obvious thatT0 + T1 ∈ V ∗0 ⊗ g0 ⊕ V ∗1 ⊗ g1 ∈ (G1 ⊗ G0)⊕ (G0 ⊗ G1),

where T0 and T1 are given by (6).

290 YUNHE SHENG

6.3. Theorem. Let (ρ0, ρ1) be a strict representation of the Lie 2-algebra G = (g0, g1, d, l2)on the 2-term complex of vector space V : V1

d−→ V0, and T0 : V0 −→ g0, T1 : V1 −→ g1linear maps. Then, (T0, T1) is an O-operator on the Lie 2-algebra G associated to therepresentation (ρ0, ρ1) if and only if T0 + T1 − σ(T0 + T1) is a solution of the 2-gradedCYBE in the semidirect product Lie 2-algebra G.Proof. It is obvious that (ρ0 ⊕ ρ1)∗ = ρ∗0 ⊕ ρ∗1 : g0 ⊕ g1 −→ gl(V ∗1 ⊕ V ∗0 ). By Lemma6.2 and Theorem 2.9, (T0, T1) is an O-operator on the Lie 2-algebra G if and only ifT0 + T1 − σ(T0 + T1) is a solution of the CYBE in the semidirect product Lie algebra(g0 n g1) nρ∗

0⊕ρ∗1

(V ∗1 ⊕ V ∗0 ), and T0 ◦ d = d ◦ T1. Note that the semidirect product Liealgebra (g0ng1)nρ∗

0⊕ρ∗1(V ∗1 ⊕V ∗0 ) is exactly the same as the semidirect product Lie algebra

G0 n G1. Furthermore, T0 ◦ d = d ◦ T1 if and only if (d ⊗ 1 − 1 ⊗ d)(T0 + T1) = 0. Thus,(T0, T1) is an O-operator on the Lie 2-algebra G associated to the representation (ρ0, ρ1)if and only if T0 + T1 − σ(T0 + T1) is a solution of the 2-graded CYBE in the semidirectproduct Lie 2-algebra G.

Let A = (A0, A1, d, ·) be a strict pre-Lie 2-algebra. Then, G(A) = (A0, A1, d, l2) is astrict Lie 2-algebra, where l2 is given by (8) and (9). Furthermore, (L0, L1) is a strictrepresentation of the Lie 2-algebra G(A) on the complex of vector spaces A1

d−→ A0,where L0, L1 are given by (11) and (12) respectively. Let {ei}1≤i≤k and {ej}1≤j≤l be thebasis of A0 and A1 respectively, and denote by {e∗i }1≤i≤k and {e∗j}1≤j≤l the dual basis.

6.4. Theorem. With the above notations,

R =k∑i=1

(ei ⊗ e∗i − e∗i ⊗ ei) +l∑

j=1(ej ⊗ e∗j − e∗j ⊗ ej) (37)

is a solution of the 2-graded CYBE in the strict Lie 2-algebra G(A) n(L∗0,L

∗1) A∗.

Proof. It is obvious that (T0 = idA0 , T1 = idA1) is an O-operator on G(A) associated tothe representation (L0, L1). By Theorem 6.3,

T0 + T1 − σ(T0 + T1) =k∑i=1

(ei ⊗ e∗i − e∗i ⊗ ei) +l∑

j=1(ej ⊗ e∗j − e∗j ⊗ ej)

is a solution of the 2-graded CYBE in the strict Lie 2-algebra G(A) n(L∗0,L

∗1) A∗.

At the end of this section, we consider the construction of strict Lie 2-bialgebras in [8,Proposition 4.4]. In fact, there are pre-Lie 2-algebras behind the construction.

Let (A, ·A) be a pre-Lie algebra. Then (A;L,R) is a representation of (A, ·A). Fur-thermore, (A∗;L∗−R∗,−R∗) is also a representation of (A, ·A). Let A0 = A and A1 = A∗.Define a multiplication · : Ai ⊗ Aj −→ Ai+j, 0 ≤ i+ j ≤ 1, by

x · y = x ·A y, x · ξ = ad∗xξ, ξ · x = −R∗xξ, ∀x, y ∈ A, ξ ∈ A∗. (38)


On the other hand, consider its sub-adjacent Lie algebra g(A). Define a skew-symmetricoperation l2 : Ai ∧ Aj −→ Ai+j, 0 ≤ i+ j ≤ 1, by

l2(x, y) = [x, y]A = x ·A y − y ·A x, l2(x, ξ) = −l2(ξ, x) = L∗xξ. (39)

6.5. Proposition. Let (A, ·A) be a pre-Lie algebra, and d : A∗ −→ A a linear map. If(A,A∗, d, ·) is a pre-Lie 2-algebra, then (g(A), A∗, d, l2) is a Lie 2-algebra, where · and l2are given by (38) and (39) respectively.

Conversely, if (g(A), A∗, d, l2) is a Lie 2-algebra, in which d : A∗ −→ A is skew-symmetric, then (A,A∗, d, ·) is a pre-Lie 2-algebra.Proof. If (A,A∗, d, ·) is a pre-Lie 2-algebra, then we have

d(ad∗xη) = x · dη, d(−R∗yξ) = (dξ) · y, ad∗dξη = −R∗dηξ, ∀x, y ∈ A, ξ, η ∈ A∗.

Therefore, we have

dl2(x, η) = dL∗xη = ad∗xη +R∗xη = x · dη − (dη) · x = l2(x, dη),l2(dξ, η) = L∗dξη = ad∗dξη +R∗dξη = ad∗dξη − ad∗dηξ = l2(ξ, dη).

Since L∗ is a representation of the Lie algebra g(A) on A∗, it is obvious that the otherconditions in the definition of a Lie 2-algebra are also satisfied. Thus, (g(A), A∗, d, l2) isa Lie 2-algebra.

Conversely, if (g(A), A∗, d, l2) is a Lie 2-algebra, we have

dl2(x, η) = l2(x, dη), l2(dξ, η) = l2(ξ, dη),

which implies thatdL∗xη = Lxdη −Rxdη, L∗dξη = −L∗dηξ.

If d is skew-symmetric, then we can obtain

〈dR∗xη, ξ〉 = 〈R∗xη,−dξ〉 = 〈η,Rxdξ〉= 〈η, Lxdξ − dL∗xξ〉 = 〈dL∗xη − Lxdη, ξ〉= 〈−Rxdη, ξ〉,

which implies thatd(η · x) = (dη) · x. (40)

Furthermore, we haved(ad∗xη) = d(L∗xη −R∗xη) = Lxdη,

which implies thatd(x · η) = x · dη. (41)

292 YUNHE SHENG

Also by the fact that d is skew-symmetric, we have

〈ad∗dξη, x〉 = 〈η, Lxdξ −Rxdξ〉 = 〈dL∗xη − dR∗xη, ξ〉= 〈Lxdη −Rxdη +Rxdη, ξ〉= 〈Rdηx, ξ〉 = 〈x,−R∗dηξ〉,

which implies that ad∗dξη = −R∗dηξ, i.e.

(dξ) · η = ξ · (dη). (42)

By (40)-(42), we deduce that Conditions (a1)-(a3) in Definition 3.2 hold. It is obviousthat the other conditions also hold. Thus, (A,A∗, d, ·) is a pre-Lie 2-algebra.

By Proposition 6.5 and Proposition 4.4 in [8], we have

6.6. Corollary. Let (A,A∗, d, ·) be a pre-Lie 2-algebra, where · is given by (38) and dis skew-symmetric. Then r given by (7) is a solution of the 2-graded CYBE in the strictLie 2-algebra (g(A), A∗, d, l2), where l2 is given by (39).Acknowledgement. The work is supported by NSFC (11471139) and NSF of JilinProvince (20170101050JC). Y. Sheng gives warmest thanks to Chengming Bai and Chen-chang Zhu for very useful comments. Y. Sheng gratefully acknowledges the support ofCourant Research Center “Higher Order Structures”, Göttingen University, where partsof this work were done during his visit.

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Department of Mathematics, Jilin University,Changchun 130012, Jilin, ChinaEmail: [email protected]

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CATEGORIFICATIONOFPRE ...CATEGORIFICATIONOFPRE-LIEALGEBRAS 271 The paper is organized as follows. In Section 2, we recall Lie 2-algebras and their representations, pre-Lie algebras